1. of 40 May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 1 Testing Affine-Invariant Properties Madhu Sudan Microsoft TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAA Surveys: works with/of Eli Ben-Sasson, Elena Grigorescu, Tali Kaufman, Shachar Lovett, Ghid Maatouk, Amir Shpilka.

2. of 40 May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 2 Property Testing … of functions from D to R: … of functions from D to R: Property P µ {D  R} Property P µ {D  R} Distance Distance δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,P) = min g 2 P [δ(f,g)] δ(f,P) = min g 2 P [δ(f,g)] f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. Local testability: Local testability: P is (k, ε, δ)-locally testable if 9 k-query test T P is (k, ε, δ)-locally testable if 9 k-query test T f 2 P ) T f accepts w.p. 1-ε. f 2 P ) T f accepts w.p. 1-ε. δ(f,P) > δ ) T f accepts w.p. ε. δ(f,P) > δ ) T f accepts w.p. ε. Notes: want k(ε, δ) = O(1) for ε,δ= (1). Notes: want k(ε, δ) = O(1) for ε,δ= (1).

3. of 40 Classical Property Test: Linearity [BLR] Does f(x+y) = f(x) + f(y), for all x, y? Does f(x+y) = f(x) + f(y), for all x, y? Variation (Affineness): Variation (Affineness): Is f(x+y) + f(0) = f(x) + f(y), for all x, y? Is f(x+y) + f(0) = f(x) + f(y), for all x, y? (roughly f(x) = a 0 +   n a i x i ) (roughly f(x) = a 0 +   n a i x i ) Test: Pick random x,y and verify above. Test: Pick random x,y and verify above. Obvious: f affine ) passes test w.p. 1. Obvious: f affine ) passes test w.p. 1. BLR Theorem: If f is δ-far from every affine function, then it fails test w.p. Ω(δ). BLR Theorem: If f is δ-far from every affine function, then it fails test w.p. Ω(δ). Ultimate goal of talk: To understand such testing results. Ultimate goal of talk: To understand such testing results. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 3

4. of 40 Affine-Invariant Properties Domain = K = GF(q n ) (field with q n elements) Domain = K = GF(q n ) (field with q n elements) Range = GF(q); q = power of prime p. Range = GF(q); q = power of prime p. P forms F-vector space. P forms F-vector space. P invariant under affine transformations of domain. P invariant under affine transformations of domain. Affine transforms? x  a.x + b, a є K *, b є K. Affine transforms? x  a.x + b, a є K *, b є K. Invariance? f є P ) g a,b (x) = f(ax+b) є P. Invariance? f є P ) g a,b (x) = f(ax+b) є P. “affine permutation of domain leaves P unchanged”. “affine permutation of domain leaves P unchanged”. Quest: What makes affine-invariant property testable? Quest: What makes affine-invariant property testable? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 4

5. of 40 (My) Goals Why? Why? BLR test has been very useful (in PCPs, LTCs). BLR test has been very useful (in PCPs, LTCs). Other derivatives equally so (low-degree test). Other derivatives equally so (low-degree test). Proof magical! Why did 3 (4) queries suffice? Proof magical! Why did 3 (4) queries suffice? Can we find other useful properties? Can we find other useful properties? Program: Program: Understand the proof better ( using invariance ). Understand the proof better ( using invariance ). Get structural understanding of affine-invariant properties, visavis local testability. Get structural understanding of affine-invariant properties, visavis local testability. Get better codes/proofs? Get better codes/proofs? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 5

6. of 40 Why’s? Why Invariance Why Invariance Natural way to abstract/unify common themes (in property testing). Natural way to abstract/unify common themes (in property testing). Graph properties, Boolean, Statistical etc.? Graph properties, Boolean, Statistical etc.? Why affine-invariance: Why affine-invariance: Abstracts linearity (affine-ness) testing. Abstracts linearity (affine-ness) testing. Low-degree testing Low-degree testing BCH testing … BCH testing … Why F-vector space? Why F-vector space? Easier to study (gives nice structure). Easier to study (gives nice structure). Common feature (in above + in codes). Common feature (in above + in codes). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 6

7. of 40 May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 7 Contrast w. Combinatorial P.T. Algebraic Property = Code! (usually) Universe Universe: {f:D  R} P Don’t care Must reject Must accept P R is a field F; P is linear!

8. of 40 Basic Implications of Linearity [BHR] If P is linear, then: If P is linear, then: Tester can be made non-adaptive. Tester can be made non-adaptive. Tester makes one-sided error Tester makes one-sided error (f 2 P ) tester always accepts). (f 2 P ) tester always accepts). Motivates: Motivates: Constraints: Constraints: k-query test => constraint of size k: k-query test => constraint of size k: value of f at ® 1,… ® k constrained to lie in subspace. value of f at ® 1,… ® k constrained to lie in subspace. Characterizations: Characterizations: If non-members of P rejected with positive probability, then P characterized by local constraints. If non-members of P rejected with positive probability, then P characterized by local constraints. functions satisfying all constraints are members of P. functions satisfying all constraints are members of P. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 8

9. of 40 f = assgm’t to left f = assgm’t to left Right = constraints Right = constraints Characterization of P: Characterization of P: P = {f sat. all constraints} P = {f sat. all constraints} Pictorially May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 9 1 2 3 D {0000,1100, 0011,1111}

10. of 40 Back to affine-invariance: More Notes Why K  F? Why K  F? Very few permutations (|K| 2 ) !! Very few permutations (|K| 2 ) !! Still “2-transitive” Still “2-transitive” Includes all properties from F n to F that are affine-invariant over F n. Includes all properties from F n to F that are affine-invariant over F n. (Hope: Maybe find a new range of parameters?) (Hope: Maybe find a new range of parameters?) Contrast with “linear-invariance” [ Bhattacharyya et al. ] Contrast with “linear-invariance” [ Bhattacharyya et al. ] Linear vs. Affine. Linear vs. Affine. Arbitrary P vs. F-vector space P Arbitrary P vs. F-vector space P Linear over F n vs. Affine over K = GF(q n ). Linear over F n vs. Affine over K = GF(q n ). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 10

11. of 40 k-local constraint k-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 11 k-locally testable k-S-O-C [KS’08]

12. of 40 Goal of this talk Definition: Single-orbit-characterization (S-O-C) Definition: Single-orbit-characterization (S-O-C) Known testable affine-invariant properties Known testable affine-invariant properties (all S-O-C!). (all S-O-C!). Structure of Affine-invariant properties. Structure of Affine-invariant properties. Non testability results Non testability results Open questions Open questions May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 12

13. of 40 Single-orbit-characterization (S-O-C) Many common properties are given by Many common properties are given by (Affine-)invariance (Affine-)invariance Single constraint. Single constraint. Example: Affineness over GF(2) n : Example: Affineness over GF(2) n : Affineness is affine-invariant. Affineness is affine-invariant. f(000000) - f(100000) ≠ f(010000) – f(110000) f(000000) - f(100000) ≠ f(010000) – f(110000) S-O-C: Abstracts this notion. S-O-C: Abstracts this notion. Suffices for testability [Kaufman+S’08] Suffices for testability [Kaufman+S’08] Unifies all known testability results!! Unifies all known testability results!! Nice structural properties. Nice structural properties. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 13

14. of 40 S-O-C: Formal Definition Constraint: Constraint: C = ( ® 1,…, ® k ;V µ F k ); ® i є K C = ( ® 1,…, ® k ;V µ F k ); ® i є K C satisfied by f if C satisfied by f if (f( ® 1 ),…,f( ® k )) є V. (f( ® 1 ),…,f( ® k )) є V. Orbit of constraint = {C o ¼ } ¼, ¼ affine. Orbit of constraint = {C o ¼ } ¼, ¼ affine. C o ¼ = ( ¼ ( ® 1 ),…, ¼ ( ® k ); V). C o ¼ = ( ¼ ( ® 1 ),…, ¼ ( ® k ); V). P has k-S-O-C, if orbit(C) characterizes P. P has k-S-O-C, if orbit(C) characterizes P. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 14

15. of 40 Known testable properties - 0 Theorem [Kaufman-S.’08]: Theorem [Kaufman-S.’08]: If P has a k-S-O-C, then P is k-locally testable. If P has a k-S-O-C, then P is k-locally testable. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 15

16. of 40 k-local constraint k-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 16 k-locally testable k-S-O-C [KS’08]

17. of 40 Known testable properties - 0 Theorem [Kaufman-S.’08]: Theorem [Kaufman-S.’08]: If P has a k-S-O-C, then P is k-locally testable. If P has a k-S-O-C, then P is k-locally testable. But who has k-S-O-C? But who has k-S-O-C? Affine functions: Affine functions: over affine transforms of F n over affine transforms of F n Degree d polynomials: Degree d polynomials: again, over affine transforms of F n again, over affine transforms of F n May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 17

18. of 40 Known testable properties - 1 Reed-Muller Property: Reed-Muller Property: View domain as F n (n-variate functions) View domain as F n (n-variate functions) Parameter d. Parameter d. RM(d) = n-var. polynomials of degree ≤ d. RM(d) = n-var. polynomials of degree ≤ d. Known to be q O(d/q) -locally testable: Known to be q O(d/q) -locally testable: Test: Test if f restricted to O(d/q)-dimensional subspace is of degree d. Test: Test if f restricted to O(d/q)-dimensional subspace is of degree d. Analysis: [Kaufman-Ron] (see appendix 1). Analysis: [Kaufman-Ron] (see appendix 1). Single-Orbit? Single-Orbit? Yes – naturally over affine transforms of F n. Yes – naturally over affine transforms of F n. Yes – unnaturally over K (field of size F n ). Yes – unnaturally over K (field of size F n ). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 18

19. of 40 Known testable properties - 2 Sparse properties: Sparse properties: Parameter t Parameter t |P| ≤ |K| t |P| ≤ |K| t Testability: Testability: Conditioned on “high-distance” [Kaufman- Litsyn, Kaufman-S.]. (no need for aff. inv.) Conditioned on “high-distance” [Kaufman- Litsyn, Kaufman-S.]. (no need for aff. inv.) Unconditionally Unconditionally [Grigorescu, Kaufman, S. ], [Kaufman- Lovett] (for prime q). [Grigorescu, Kaufman, S. ], [Kaufman- Lovett] (for prime q). Also S-O-C. Also S-O-C. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 19

20. of 40 Known Testable Properties - 3 Intersections: Intersections: P 1  P 2 always locally testable, also S-O-C. P 1  P 2 always locally testable, also S-O-C. Sums: Sums: P 1 + P 2 (= {f 1 + f 2 | f i є P i }) P 1 + P 2 (= {f 1 + f 2 | f i є P i }) S-O-C iff P 1 and P 2 are S-O-C [BGMSS’11] S-O-C iff P 1 and P 2 are S-O-C [BGMSS’11] Lifts [BMSS’11] Lifts [BMSS’11] Suppose F µ L µ K. Suppose F µ L µ K. P µ {L  F} has k-S-O-C, with constraint C. P µ {L  F} has k-S-O-C, with constraint C. Then Lift_{L  K}(P) = property characterized by K-orbit(C). Then Lift_{L  K}(P) = property characterized by K-orbit(C). By Definition: Lift(P) is k-S-O-C. By Definition: Lift(P) is k-S-O-C. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 20

21. of 40 Known Testable Properties - 1 Finite combination of Lifts, Intersections, Sums of Sparse and Reed-Muller properties. Finite combination of Lifts, Intersections, Sums of Sparse and Reed-Muller properties. Known: They are testable (for prime q). Known: They are testable (for prime q). Open: Are they the only testable properties? Open: Are they the only testable properties? If so, Testability ≡ Single-Orbit. If so, Testability ≡ Single-Orbit. First target: n = prime: First target: n = prime: no lifts/intersections; only need to show that every testable property is sum of sparse and Reed-Muller property. no lifts/intersections; only need to show that every testable property is sum of sparse and Reed-Muller property. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 21

22. of 40 May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 22 Affine-Invariant Properties: Structure

23. of 40 Preliminaries Every function from K  K, including K  F, Every function from K  K, including K  F, is a polynomial in K[x] is a polynomial in K[x] So every property P = {set of polynomials}. So every property P = {set of polynomials}. Is set arbitrary? Any structure? Is set arbitrary? Any structure? Alternate representation: Alternate representation: Tr(x) = x + x q + x q 2 + … + x q n-1 Tr(x) = x + x q + x q 2 + … + x q n-1 Tr(x+y) = Tr(x)+Tr(y); Tr( ® x) = ® Tr(x), ® є F. Tr(x+y) = Tr(x)+Tr(y); Tr( ® x) = ® Tr(x), ® є F. Tr: K  F. Tr: K  F. Every function from K  F is Tr(f) for some polynomial f є K[x]. Every function from K  F is Tr(f) for some polynomial f є K[x]. Any structure to these polynomials? Any structure to these polynomials? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 23

24. of 40 Example F = GF(2), K = GF(2 n ). F = GF(2), K = GF(2 n ). Suppose P contains Tr(x 11 + x 3 + 1). Suppose P contains Tr(x 11 + x 3 + 1). What other functions must P contain (to be affine-invariant)? What other functions must P contain (to be affine-invariant)? Claims: Claims: Let D = {0,1,3,5,9,11}. Let D = {0,1,3,5,9,11}. Then P contains every function of the form Tr(f), where f is supported on monomials with degrees from D. Then P contains every function of the form Tr(f), where f is supported on monomials with degrees from D. So Tr(x 5 ),Tr( ® x 9 + ¯ x 5 ),Tr(x 11 +x 5 +x 3 +x)+1 є P. So Tr(x 5 ),Tr( ® x 9 + ¯ x 5 ),Tr(x 11 +x 5 +x 3 +x)+1 є P. How? Why? How? Why? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 24

25. of 40 Structure - 1 Definitions: Definitions: Deg(P) = {d | 9 f є P, with x d є supp(f)} Deg(P) = {d | 9 f є P, with x d є supp(f)} Fam(D) = {f: K  F | supp(f) µ D} Fam(D) = {f: K  F | supp(f) µ D} Proposition: For affine-invariant property P Proposition: For affine-invariant property P P = Fam(Deg(P)). P = Fam(Deg(P)). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 25

26. of 40 Structure - 2 Definitions: Definitions: Shift(d) = {d, q.d, q 2.d, … } mod (q n -1). Shift(d) = {d, q.d, q 2.d, … } mod (q n -1). D is shift-closed if Shift(D) = D. D is shift-closed if Shift(D) = D. e ≤ d : e = e 0 + e 1 p + …; e ≤ d : e = e 0 + e 1 p + …; d = d 0 + d 1 p + …; d = d 0 + d 1 p + …; e ≤ d if e i ≤ d i for all i. e ≤ d if e i ≤ d i for all i. Shadow(d) = {e ≤ d}; Shadow(d) = {e ≤ d}; Shadow(D) = [ d є D Shadow(d). Shadow(D) = [ d є D Shadow(d). D is shadow-closed if Shadow(D) = D. D is shadow-closed if Shadow(D) = D. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 26

27. of 40 Structure - 3 Proposition: For every affine-invariant property P, Deg(P) is p-shadow-closed and q-shift-closed. Proposition: For every affine-invariant property P, Deg(P) is p-shadow-closed and q-shift-closed. (Shadowing comes from affine-transforms; Shifts come from range being F). Proposition: For every p-shadow-closed, q-shift- closed family D, Fam(D) is affine-invariant and Proposition: For every p-shadow-closed, q-shift- closed family D, Fam(D) is affine-invariant and D = Deg(Fam(D)) May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 27

28. of 40 Example revisited Tr(x 11 + x 3 ) є P Tr(x 11 + x 3 ) є P Deg(P)  11, 3 (definition of Deg) Deg(P)  11, 3 (definition of Deg) Deg(P)  11, 9, 5, 3, 1, 0 (shadow-closure) Deg(P)  11, 9, 5, 3, 1, 0 (shadow-closure) Deg(P)  Tr(x 11 ), Tr(x 9 ) etc. (shift-closure). Deg(P)  Tr(x 11 ), Tr(x 9 ) etc. (shift-closure). Fam(Deg(P))  Tr(x 11 ) etc. (definition of Fam). Fam(Deg(P))  Tr(x 11 ) etc. (definition of Fam). P  Tr(x 11 ) (P = Fam(Deg(P))) P  Tr(x 11 ) (P = Fam(Deg(P))) May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 28

29. of 40 What kind of properties have k-S-O-C? (Positive results interpreted structurally) If propery has all degrees of q-weight at most k then it is RM and has (q k )-S-O-C: If propery has all degrees of q-weight at most k then it is RM and has (q k )-S-O-C: q-weight(d) =   d i, q-weight(d) =   d i, where d = d 0 + d 1 q + … where d = d 0 + d 1 q + … Also, if P = Fam(D) & D = Shift(S) for small shadow-closed S, then P is k(|S|)-S-O-C. Also, if P = Fam(D) & D = Shift(S) for small shadow-closed S, then P is k(|S|)-S-O-C. (Alternate definition of sparsity.) (Alternate definition of sparsity.) Other examples from Intersection, Sum, Lift. Other examples from Intersection, Sum, Lift. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 29

30. of 40 What affine-invariant properties are not locally testable. Very little known. Very little known. Specific examples: Specific examples: GKS08: Exists a-i property with k-local constraint which is not k-locally characterized. GKS08: Exists a-i property with k-local constraint which is not k-locally characterized. BMSS11: Exists k-locally characterized a-i property that is not testable. BMSS11: Exists k-locally characterized a-i property that is not testable. BSS’10: If wt(d) ¸ k for some d in Deg(P), then P does not have a k-local constraint. BSS’10: If wt(d) ¸ k for some d in Deg(P), then P does not have a k-local constraint. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 30

31. of 40 [BS’10]k-local constraint k-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 31 k-locally testable k-S-O-C [KS’08] [GKS’08] [BMSS’11]weight-k degrees

32. of 40 Quest in lower bound Given degree set D (shadow-closed, shift-closed) prove it has no S-O-C. Given degree set D (shadow-closed, shift-closed) prove it has no S-O-C. Equivalently: Prove there are no Equivalently: Prove there are no ¸ 1 … ¸ k є F, ® 1 … ® k є K such that ¸ 1 … ¸ k є F, ® 1 … ® k є K such that  i=1 k ¸ i ® i d = 0 for every d є D.  i=1 k ¸ i ® i d = 0 for every d є D.  i=1 k ¸ i ® i d ≠ 0 for every minimal d  D.  i=1 k ¸ i ® i d ≠ 0 for every minimal d  D. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 32

33. of 40 Pictorially May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 33 ®1d®1d ®2d®2d ®kd®kd … M(D) = Is there a vector ( ¸ 1,…, ¸ k ) in its right kernel? Can try to prove “NO” by proving matrix has full rank. Unfortunately, few techniques to prove non-square matrix has high rank.

34. of 40 Non-testable Property - 1 AKKLR ( Alon,Kaufman,Krivelevich,Litsyn,Ron ) Conjecture: AKKLR ( Alon,Kaufman,Krivelevich,Litsyn,Ron ) Conjecture: If a linear property is 2-transitive and has a k- local constraint then it is testable. If a linear property is 2-transitive and has a k- local constraint then it is testable. [GKS’08]: For every k, there exists affine- invariant property with 8-local constraint that is not k-locally testable. [GKS’08]: For every k, there exists affine- invariant property with 8-local constraint that is not k-locally testable. P = Fam(Shift({0,1} [ {1+2,1+2 2,…,1+2 k })). P = Fam(Shift({0,1} [ {1+2,1+2 2,…,1+2 k })). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 34

35. of 40 Proof (based on [BMSS’11]) F = GF(2); K = GF(2 n ); F = GF(2); K = GF(2 n ); P k = Fam(Shift({0,1} [ {1 + 2 i | i є {1,…,k}})) P k = Fam(Shift({0,1} [ {1 + 2 i | i є {1,…,k}})) Let M i = Let M i = If Ker(M i ) = Ker(M i+1 ), then Ker(M i+2 ) = Ker(M i ) If Ker(M i ) = Ker(M i+1 ), then Ker(M i+2 ) = Ker(M i ) Ker(M k+1 ) = would accept all functions in P k+1 Ker(M k+1 ) = would accept all functions in P k+1 So Ker(M i ) must go down at each step, implying Rank(M_{i+1}) > Rank(M_i). So Ker(M i ) must go down at each step, implying Rank(M_{i+1}) > Rank(M_i). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 35 ®12i®12i … ®12®12 ®22®22 ®k2®k2 … ®22i®22i ®k2i®k2i

36. of 40 Stronger Counterexample GKS counterexample: GKS counterexample: Takes AKKLR question too literally; Takes AKKLR question too literally; Of course, a non-locally-characterizable property can not be locally tested. Of course, a non-locally-characterizable property can not be locally tested. Weaker conjecture: Weaker conjecture: Every k-locally characterized affine-invariant (2-transitive) property is locally testable. Every k-locally characterized affine-invariant (2-transitive) property is locally testable. Alas, not true: [BMSS] Alas, not true: [BMSS] May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 36

37. of 40 [BMSS] CounterExample Recall: Recall: Every known locally characterized property was locally testable Every known locally characterized property was locally testable Every known locally testable property is S-O-C. Every known locally testable property is S-O-C. Need a locally characterized property which is (provably) not S-O-C. Need a locally characterized property which is (provably) not S-O-C. Idea: Idea: Start with sparse family P i. Start with sparse family P i. Lift it to get Q i (still S-O-C). Lift it to get Q i (still S-O-C). Take intersection of superconstantly many such properties. Q =  i Q i Take intersection of superconstantly many such properties. Q =  i Q i May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 37

38. of 40 Example: Sums of S-O-C properties Suppose D 1 = Deg(P 1 ) and D 2 = Deg(P 2 ) Suppose D 1 = Deg(P 1 ) and D 2 = Deg(P 2 ) Then Deg(P 1 + P 2 ) = D 1 [ D 2. Then Deg(P 1 + P 2 ) = D 1 [ D 2. Suppose S-O-C of P 1 is C 1 : f(a 1 ) + … + f(a k ) = 0; and S-O-C of P 2 is C 2 : f(b 1 ) + … + f(b k ) = 0. Suppose S-O-C of P 1 is C 1 : f(a 1 ) + … + f(a k ) = 0; and S-O-C of P 2 is C 2 : f(b 1 ) + … + f(b k ) = 0. Then every g є P 1 + P 2 satisfies: Then every g є P 1 + P 2 satisfies:  i,j g(a i b j ) = 0  i,j g(a i b j ) = 0 Doesn’t yield S-O-C, but applied to random constraints in orbit(C 1 ), orbit(C 2 ) does! Doesn’t yield S-O-C, but applied to random constraints in orbit(C 1 ), orbit(C 2 ) does! Proof uses wt(Deg(P 1 )) ≤ k. Proof uses wt(Deg(P 1 )) ≤ k. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 38

39. of 40 Concluding Affine-invariance gives nice umbrella to capture algebraic property testing: Affine-invariance gives nice umbrella to capture algebraic property testing: Important (historically) for PCPs, LTCs, LDCs. Important (historically) for PCPs, LTCs, LDCs. Incorporates symmetry. Incorporates symmetry. Would be nice to have a complete characterization of testability of affine-invariant properties. Would be nice to have a complete characterization of testability of affine-invariant properties. Understanding (severely) lacking. Understanding (severely) lacking. Know: Know: Can’t be much better than Reed-Muller. Can’t be much better than Reed-Muller. Can they be slightly better? YES! Can they be slightly better? YES! May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 39

40. of 40 Thank You! May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 40

41. of 40 Appendix 0: Main References Early results mentioning invariance: Early results mentioning invariance: [Babai, Shpilka, Stefankovic] Cyclic codes [Babai, Shpilka, Stefankovic] Cyclic codes [AKKLR] 2-transitivity [AKKLR] 2-transitivity [Goldreich-Sheffett] Lower bounds on randomness required. [Goldreich-Sheffett] Lower bounds on randomness required. Affine-invariance Affine-invariance [Kaufman-Sudan, STOC ‘08] [Kaufman-Sudan, STOC ‘08] [Grigorescu, Kaufman, Sudan, CCC ‘08] [Grigorescu, Kaufman, Sudan, CCC ‘08] [Grigorescu, Kaufman, Sudan, Random ‘09] [Grigorescu, Kaufman, Sudan, Random ‘09] [Kaufman, Lovett, ECCC TR10-065] [Kaufman, Lovett, ECCC TR10-065] [Ben-Sasson, Sudan, ECCC TR 10-108] [Ben-Sasson, Sudan, ECCC TR 10-108] [Ben-Sasson, Maatouk, Shpilka, Sudan, CCC‘11] [Ben-Sasson, Maatouk, Shpilka, Sudan, CCC‘11] [Ben-Sasson, Grigorescu, Maatouk, Shpilka, Sudan, ECCC TR 11-079] [Ben-Sasson, Grigorescu, Maatouk, Shpilka, Sudan, ECCC TR 11-079] Other related themes: Other related themes: Fair amount of work on (non-linear, linear-invariance) – refs omitted. Fair amount of work on (non-linear, linear-invariance) – refs omitted. Kaufman+Wigderson, ??? – other algebraic invariances (1-transitive) Kaufman+Wigderson, ??? – other algebraic invariances (1-transitive) Goldreich + Kaufman – general relationships between invariance and testing Goldreich + Kaufman – general relationships between invariance and testing May 23-28, 2011 Bertinor: Testing Affine-Invariant Properties 41

42. of 40 Appendix 1 Reed-Muller testing: Reed-Muller testing: Early works (PCP etc.): consider only d < q. Early works (PCP etc.): consider only d < q. [Rubinfeld, Sudan ‘92] [Rubinfeld, Sudan ‘92] [Arora, Safra ’92] [Arora, Safra ’92] [ALMSS ‘92] [ALMSS ‘92] Variations (multilinear, low indiv. degree) due to [BFL,BFLS,FGLSS]. Variations (multilinear, low indiv. degree) due to [BFL,BFLS,FGLSS]. d > n: q d > n: q q = 2: [Alon,Kaufman,Krivelevich,Litsyn,Ron ’02] q = 2: [Alon,Kaufman,Krivelevich,Litsyn,Ron ’02] general q: [Kaufman Ron ’04] general q: [Kaufman Ron ’04] (prime q): [Jutla Patthak Rudra Zuckerman ‘04.] (prime q): [Jutla Patthak Rudra Zuckerman ‘04.] Tight results: Tight results: q=2: O(2^d)-locally testable: q=2: O(2^d)-locally testable: [Bhattacharyya,Kopparty,Schoenebeck,Sudan,Zuckerman ‘09] [Bhattacharyya,Kopparty,Schoenebeck,Sudan,Zuckerman ‘09] general q: q^{d/(q-q/p} [Haramaty Shpilka Sudan ’11] general q: q^{d/(q-q/p} [Haramaty Shpilka Sudan ’11] tight for prime q; open for general q. tight for prime q; open for general q. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 42

43. of 40 Appendix 2: Analysis of k-S-O-C test Property P (k-S-O-C) given by ® 1,…, ® k ; V 2 F k Property P (k-S-O-C) given by ® 1,…, ® k ; V 2 F k P = {f | f(A( ® 1 )) … f(A( ® k )) 2 V, 8 affine A:K n K n } P = {f | f(A( ® 1 )) … f(A( ® k )) 2 V, 8 affine A:K n K n } Rej(f) = Prob A [ f(A( ® 1 )) … f(A( ® k )) not in V ] Rej(f) = Prob A [ f(A( ® 1 )) … f(A( ® k )) not in V ] Wish to show: If Rej(f) < 1/k 3, Wish to show: If Rej(f) < 1/k 3, then δ(f,P) = O(Rej(f)). then δ(f,P) = O(Rej(f)). May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 43

44. of 40 Appendix 2: BLR Analog Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Define g(x) = majority y {Vote x (y)}, Define g(x) = majority y {Vote x (y)}, where Vote x (y) = f(x+y) – f(y). where Vote x (y) = f(x+y) – f(y). Step 0: Show δ(f,g) small Step 0: Show δ(f,g) small Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 2: Use above to show g is well-defined and a homomorphism. Step 2: Use above to show g is well-defined and a homomorphism. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 44

45. of 40 Appendix 2: BLR Analysis of Step 1 Why is f(x+y) – f(y) = f(x+z) – f(z), usually? Why is f(x+y) – f(y) = f(x+z) – f(z), usually? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 45 - f(x+z) f(y) - f(x+y) f(z) -f(y) f(x+y+z) -f(z) 0 ?

46. of 40 Appendix 2: Generalization g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, Pr A [ ¯,f(A( ® 2 ),…,f(A( ® k )) 2 V] Pr A [ ¯,f(A( ® 2 ),…,f(A( ® k )) 2 V] Step 0: δ(f,g) small. Step 0: δ(f,g) small. Vote x (A) = ¯ s.t. ¯, f(A( ® 2 ))…f(A( ® k )) 2 V Vote x (A) = ¯ s.t. ¯, f(A( ® 2 ))…f(A( ® k )) 2 V (if such ¯ exists) (if such ¯ exists) Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 2: Use above to show g 2 P. Step 2: Use above to show g 2 P. May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 46

47. of 40 Appendix 2: Matrix Magic? May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 47 A( ® 2 ) B( ® k ) B( ® 2 ) A( ® k ) x t Say A( ® 1 ) … A( ® t ) independent; rest dependent t Random No Choice Doesn’t Matter!